The simulation results of ER networks with 20000 nodes for different <i>β</i>.

<p>In the simulation, the parameters are set as: ⟨<i>k</i>⟩ = 6, <i>g</i> = 5. (a) The size of the giant component <i>S</i> versus <i>p</i>, the fraction of nodes that have been left after random removal. The symbols represent simulation results, and the solid lines show the corresponding analytical predictions of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126674#pone.0126674.e010" target="_blank">Eq (9)</a>. For <i>β</i> > (<i>g</i> − 1)/<i>g</i>, the dependence group has no effect on the cascading process, then the percolation process leads to a second order phase transition. When <i>β</i> < (<i>g</i> − 1)/<i>g</i>, a first order phase transition can be found. Both the first and the second order phase transition processes obey <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126674#pone.0126674.e010" target="_blank">Eq (9)</a>. (b) As pointed in ref.[<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126674#pone.0126674.ref010" target="_blank">10</a>], the number of iterative failures (NOI) sharply increases when <i>p</i> approaches the critical point, so the sharp peaks can identify the corresponding critical points </p><p></p><p></p><p><mi>p</mi><mi>c</mi><mi>I</mi></p><p></p><p></p> for the first order transition region.<p>